Homological properties of modules with finite weak injective and weak flat dimensions

Gorenstein homological theory for differential modules

We show that a differential module is Gorenstein projective if and only if its underlying module is Gorenstein projective. Dually, a differential module is Gorenstein injective if and only if its underlying module is Gorenstein injective.

Relative FP-injective and FP-flat complexes and their model structures

In this paper, we introduce the notions of ${\rm FP}_n$-injective and ${\rm FP}_n$-flat complexes in terms of complexes of type ${\rm FP}_n$. We show that some characterizations analogous to that of injective, FP-injective and flat complexes exist for ${\rm FP}_n$-injective and ${\rm FP}_n$-flat complexes. We also introduce and study ${\rm FP}_n$-injective and ${\rm FP}_n$-flat dimensions of modules and complexes, and give a relation between them in terms of Pontrjagin duality. The existence of pre-envelopes and covers in this setting is discussed, and we prove that any complex has an ${\rm FP…

Canonical Filtrations of Gorenstein Injective Modules

The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So perhaps it is surprising that tensor products of Gorenstein injective modules over Gorenstein rings of finite Krull dimension are Gorenstein injective. Our main result is in support of the princ…

Homological aspects of semidualizing modules

We investigate the notion of the C-projective dimension of a module, where C is a semidualizing module. When C=R, this recovers the standard projective dimension. We show that three natural definitions of finite C-projective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results that demonstrate the deep connections between modules of finite projective dimension and modules of finite C-projective dimension. In parallel, we develop the dual theory for injective dimension and C-injec…

Gorenstein homological dimensions of modules over triangular matrix rings

Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=\left(\begin{smallmatrix} A & 0 \\ U & B \\\end{smallmatrix}\right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over $T$, and discuss when a left $T$-module is strongly Gorenstein projective or strongly Gorenstein injective module.

Remarks on Gorenstein weak injective and weak flat modules

In this paper, we introduce the notions of Gorenstein weak injective and weak flat modules respectively in terms of weak injective and weak flat modules, which is larger than classical classes of Gorenstein injective and flat modules. In this new setting, we characterize rings over which all modules are Gorenstein weak injective. Moreover, we also discuss a relation between weak cosyzygy and Gorenstein weak cosyzygy of a module, and the stability of Gorenstein weak injective modules.

On Gorenstein Projective, Injective and Flat Dimensions - A Functorial Description with Applications

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: It now includes, for instance, the rings encountered in commutative algebraic geometry and, in the…

Injective Envelopes and (Gorenstein) Flat Covers

We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring $R$, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left $R$-module is at most the flat dimension of the injective envelope of $_RR$. Then we get that the injective envelope of $_RR$ is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left $R$-module is (Gorenstein) flat, if and only if the injective envelope of every flat left $R$-module is (Gorenstein) flat, if and only if…

温和代数的同调界

Homological Bounds of Gentle algebras

本文研究温和代数上不可分解模的投射维数与内射维数之和的上界，给出该和严格小于整体维数两倍的条件，并由此刻画拟倾斜温和代数。

Bounds on homological invariants of VI-modules

We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.

正交模的同调理论

Homological theory of orthogonal modules

本研究从稳定模范畴视角研究自入射Artin代数上的有限生成正交生成元，建立M-相对稳定范畴的粘合、刻画紧对象，给出Tachikawa第二猜想的等价刻画，还引入Gorenstein-Morita代数并证明Nakayama猜想对其成立。

导出函子、解析与n-ary伽马半环的同调对偶性

Derived Functors, Resolutions, and Homological Dualities in n-ary Gamma-Semirings

本文研究了非交换n-ary伽马半环的同调结构，定义了双伽马模的导出函子，并构建了相关谱序列和几何分析框架。

Ore localization and minimal injective resolutions

In this paper, we describe the structure of the localization of Ext^{i}_{R}(R/P,M), where P is a prime ideal and M is a module, at certain Ore sets X. We first study the situation for FBN rings, and then consider matters for a large class of Auslander-Gorenstein rings. We need to impose suitable homological regularity conditions to get results in the more general situation. The results obtained are then used to study the shape of minimal injective resolutions of modules over noetherian rings.

Retractions and Gorenstein homological properties

We associate to a localizable module a left retraction of algebras; it is a homological ring epimorphism that preserves singularity categories. We study the behavior of left retractions with respect to Gorenstein homological properties (for example, being Gorenstein algebras or CM-free). We apply the results to Nakayama algebras. It turns out that for a connected Nakayama algebra $A$, there exists a connected self-injective Nakayama algebra $A'$ such that there is a sequence of left retractions linking $A$ to $A'$; in particular, the singularity category of $A$ is triangle equivalent to the st…

Homological behavior of Auslander's $k$-Gorenstein rings

In this paper we mainly study the homological properties of dual modules over $k$-Gorenstein rings. For a right quasi $k$-Gorenstein ring $Λ$, we show that the right self-injective dimension of $Λ$ is at most $k$ if and only if each $M \in$mod $Λ$ satisfying the condition that Ext$_Λ^i(M, Λ)=0$ for any $1\leq i \leq k$ is reflexive. For an $\infty$-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of $Λ$ are identical. In addition, we show that if $Λ$ is a left quasi $\infty$-Gorenstein ring and $M\in$mod $Λ$ with grade$M$ finite, then Ext$_…

诺特代数上内射包络与平坦覆盖的对偶性

Duality Between Injective Envelopes and Flat Covers over Noether Algebras

本文将交换诺特环上内射包络与平坦覆盖的对偶关系推广到非交换诺特代数领域，拓展了相关同调代数结论的适用范围。

Absolutely Clean, Level, and Gorenstein AC-Injective Complexes

Absolutely clean and level $R$-modules were introduced in [BGH13] and used to show how Gorenstein homological algebra can be extended to an arbitrary ring $R$. This led to the notion of Gorenstein AC-injective and Gorenstein AC-projective $R$-modules. Here we study these concepts in the category of chain complexes of $R$-modules. We define, characterize and deduce properties of absolutely clean, level, Gorenstein AC-injective, and Gorenstein AC-projective chain complexes. We show that the category $\text{Ch}(R)$ of chain complexes has a cofibrantly generated model structure where every object …

Foxby equivalence relative to $C$-weak injective and $C$-weak flat modules

Let $S$ and $R$ be rings and $_SC_R$ a (faithfully) semidualizing bimodule. We introduce and study $C$-weak flat and $C$-weak injective modules as a generalization of $C$-flat and $C$-injective modules (J. Math. Kyoto Univ. 47(2007), 781-808) respectively, and use them to provide additional information concerning the important Foxby equivalence between the subclasses of the Auslander class $\mathcal{A}_C(R)$ and that of the Bass class $\mathcal{B}_C(S)$. Then we study the stability of Auslander and Bass classes, which enables us to give some alternative characterizations of the modules in $\ma…

Rigidity dimension - a homological dimension measuring resolutions of algebras by algebras of finite global dimension

A new homological dimension is introduced to measure the quality of resolutions of `singular' finite dimensional algebras (of infinite global dimension) by `regular' ones (of finite global dimension). Upper bounds are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological conjectures. Then invariance under stable equivalences is shown to hold, with some exceptions when there are nodes in case of additive equivalences, and without exceptions in case of triangulated equivalences. Stable equivalences of Morita type and derived equiva…

Torsion pairs over $n$-Hereditary rings

We study the notions of $n$-hereditary rings and its connection to the classes of finitely $n$-presented modules, FP$_n$-injective modules, FP$_n$-flat modules and $n$-coherent rings. We give characterizations of $n$-hereditary rings in terms of quotients of injective modules and submodules of flat modules, and a characterization of $n$-coherent using an injective cogenerator of the category of modules. We show two torsion pairs with respect to the FP$_n$-injective modules and the FP$_n$-flat modules over $n$-hereditary rings. We also provide an example of a Bézout ring which is 2-hereditary, …

Relatively divisible and relatively flat objects in exact categories

We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair $(\mathcal{A},\mathcal{B})$ in an exact category $\mathcal{C}$, $\mathcal{A}$ coincides with the class of relatively flat objects of $\mathcal{C}$ for some relative projectively generated exact structure, while $\mathcal{B}$ coincides with the class of relatively divisible objects of $\mathcal{C}$ for some relative injectively generated exact structure. We exhibit Galois connections between relative cotorsion pairs, relative projectively generat…

Acyclic complexes and Gorenstein rings

For a given class of modules $\mathcal{A}$, we denote by $\widetilde{\mathcal{A}}$ the class of exact complexes $X$ having all cycles in $\mathcal{A}$, and by $dw(\mathcal{A})$ the class of complexes $Y$ with all components $Y_j$ in $\mathcal{A}$. We use the notations $\mathcal{GI}$ $(\mathcal{GF}, \mathcal{GP})$ for the class of Gorenstein injective (Gorenstein flat, Gorenstein projective respectively) $R$-modules, $\mathcal{DI}$ for Ding injective modules, and $\mathcal{PGF}$ for projectively coresolved Gorenstein flat modules (see section 2 for definitions). We prove that the following are …

RD-flatness and RD-injectivity

It is proved that every commutative ring whose RD-injective modules are $Σ$-RD-injective is the product of a pure semi-simple ring and a finite ring. A complete characterization of commutative rings for which each artinian (respectively simple) module is RD-injective, is given. These results can be obtained by using the properties of RD-flat modules and RD-coflat modules which are respectively the RD-relativization of flat modules and fp-injective modules. It is also shown that a commutative ring is perfect if and only if each RD-flat module is RD-projective.

Category of n-weak injective and n-weak flat modules with respect to special super presented modules

Let $R$ be a ring and $n$, $k$ two non-negative integers. In this paper, we introduce the concepts of $n$-weak injective and $n$-weak flat modules and via the notion of special super finitely presented modules, we obtain some characterizations of these modules. We also investigate two classes of modules with richer contents, namely $\mathcal{WI}_k^n(R)$ and $\mathcal{WF}_k^n(R^{op})$ which are larger than that of modules with weak injective and weak flat dimensions less than or equal to $k$. Then on any arbitrary ring, we study the existence of $\mathcal{WI}_k^n(R)$ and $\mathcal{WF}_k^n(R^{op…

$\rm{FP}_{n}$-injective and $\rm{FP}_{n}$-flat covers and preenvelopes, and Gorenstein AC-flat covers

We prove that, for any $n \geq 2$, the classes of $\rm{FP}_{n}$-injective modules and of $\rm{FP}_n$-flat modules are both covering and preenveloping over any ring $R$. This includes the case of $\rm{FP}_{\infty}$-injective and $\rm{FP}_{\infty}$-flat modules (i.e. absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules - the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gor…

(Strongly) $M-\pazocal{A}-$Injective(Flat) Modules

Let $M$ be a left $R-$module and $\pazocal{A}=\{A\}_{A\in\pazocal{A}}$ be a family of some submodules of $M$. It is introduced the classes of (strongly) $M-\pazocal{A}-\mathrm{injective}$ and (strongly) $M-\pazocal{A}-\mathrm{flat}$ modules which are denoted by $(S) M-\pazocal{A}I$ and $(S) M-\pazocal{A}F$, respectively. It is obtained some characterizations of these classes and the relationships between these classes. Moreover it is investigated $(S) M-\pazocal{A}I$ and $(S) M-\pazocal{A}F$ precovers and preenvelopes of modules. It is also studied $\pazocal{A}$-coherent, $F\pazocal{A}$ and $P…

Homological properties of Orlik-Solomon algebras

The Orlik-Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first we study homological properties of E-modules as e.g. complexity, depth and regularity. In particular, we consider modules with linear injective resolutions. We apply our results to Orlik-Solomon algebras of matroids and give formulas for the complexity, depth and regularity of such rings in terms of invariants of the matroid. Moreover, we characterize those matroids whose Orlik-Solomon ideal has a linear projective resolution and compute in these cases the Betti numbers of the idea…

平凡环扩张上的Gorenstein投射、内射和平坦模

Gorenstein projective, injective and flat modules over trivial ring extensions

本文通过引入广义相容与广义余相容双模，给出了平凡环扩张上Gorenstein投射、内射和平坦模的显式刻画，并应用于具有零双模同态的Morita context环。

Homological dimension based on a class of Gorenstein flat modules

In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying …

Homological Aspects of the Dual Auslander Transpose II

Let $R$ and $S$ be rings and $_Rω_S$ a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of $\infty$-$ω$-cotorsionfree modules and a subclass of the class of $ω$-adstatic modules. Also we establish the relation between the relative homological dimensions of a module $M$ and the corresponding standard homological dimensions of ${\rm Hom}(ω,M)$. By investigating the properties of the Bass injective dimension of modules (resp. complexes), we get some equivalent characterizations of semi-tilting modules (resp. Gorenstein artin algebras). Finally we obtain a …

通过函子范畴研究具有 Grothendieck 心的 $t$-结构

$t$-Structures with Grothendieck hearts via functor categories

研究带余积三角范畴中 $t$-结构的心成为 AB5 或 Grothendieck 范畴的条件，通过函子范畴将心表示为 Serre 商并分类保余积同调函子，证明标准良生成三角范畴中纯内射对象总余生成 $t$-结构。

Gorenstein derived functors from special rings to arbitrary rings

In this note, we mainly extend some Gorenstein homological properties from special rings (Noetherian or coherent rings ) to arbitrary rings by introducing the notions of Gorenstein weak injective and weak projective modules respectively.

$(\mathcal{F},\mathcal{A})$-Gorenstein flat homological dimensions

In this paper we develop the homological properties of the $(\mathcal{L}, \mathcal{A})$-Gorenstein flat $R$-modules $\mathcal{GF}_{(\mathcal{F}(R), \mathcal{A})}$ proposed by Gillespie. Where the class $\mathcal{A} \subseteq \mathrm{Mod} (R^{op})$ sometimes corresponds to a duality pair $(\mathcal{L}, \mathcal{A})$. We study the weak global and finitistic dimensions that comes with $\mathcal{GF}_{(\mathcal{F}(R), \mathcal{A})}$ and show that over a $(\mathcal{L}, \mathcal{A})$-Gorenstein ring, the functor $-\otimes _R -$ is left balanced over $\mathrm{Mod} (R^{op}) \times \mathrm{Mod} (R)$ by …

Generalized Gorensteinness and a homological determinant for preprojective algebras

The study of invariants of group actions on commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. It has been of particular interest to understand what conditions on the group result in an invariant ring satisfying useful properties. In particular, Watanabe's Theorem states that the invariant subring of $k[x_1,\ldots,x_n]$ under the natural action of a finite subgroup of $SL_n(k)$ is always Gorenstein. In this paper, we study this question in the more general setting of group actions on noncommutative non-connected algebras $A$. We develop …

非交换n元Gamma半环的精确范畴与同调基础

Exact Categories and Homological Foundations of Non-Commutative n-ary Gamma. Semirings

本文提出非交换n元Gamma半环的同调基础，构建了非交换衍生Gamma几何框架，实现了Gamma代数中两个不同方向的统一。

特殊半对偶双模下的$(n,d)$-内射模与$(n,d)$-平坦模

$(n,d)$-injective and $(n,d)$-flat modules under a special semidualizing bimodule

本文在特殊半对偶双模框架下引入并研究$K_{d-1}$-$(n,d)$-内射模与$K_{d-1}$-$(n,d)$-平坦模，统一推广了多类已知模，证明了相应模类的覆盖性与预包络性，并研究了Foxby等价及$n$-凝聚环上的封闭性质。

On Auslander-Type Conditions of Modules

We prove that for a left and right Noetherian ring $R$, $_RR$ satisfies the Auslander condition if and only if so does every flat left $R$-module, if and only if the injective dimension of the $i$th term in a minimal flat resolution of any injective left $R$-module is at most $i-1$ for any $i \geq 1$, if and only if the flat (resp. injective) dimension of the $i$th term in a minimal injective coresolution (resp. flat resolution) of any left $R$-module $M$ is at most the flat (resp. injective) dimension of $M$ plus $i-1$ for any $i \geq 1$, if and only if the flat (resp. injective) dimension of…

Localization of injective modules over valuation rings

It is proved that the localization of an injective module E, over a valuation ring R, at a prime ideal J, is injective if J is not the subset of zero-divisors of R or if J or E is flat. It follows that localizations of injective modules over h-local Prüfer domains are injective too.

n-gr-Coherent rings and Gorenstein graded modules

Let R be a graded ring and n > 1 an integer. In this paper, We introduce the notions of Ding n-gr-injective and Ding n-gr-flat modules by using of special finitely presented graded modules . Then, some properties of Ding n-gr-injective and Ding n-gr-flat modules are obtained. On n-gr-coherent rings, we investigate the relationships among Ding n-gr-injective and Ding n-gr-flat modules and also, we prove that any graded module in R-gr (resp. gr-R) admits an Ding n-gr-injective (resp. Ding n-gr-flat) cover and preenvelope.

关于 $C$-$fp_n$-内射与 $C$-$fp_{n}$-平坦模的 Foxby 等价

Foxby equivalence relative to $C$-$fp_n$-injective and $C$-$fp_{n}$-flat modules

本文引入 $C$-$fp_n$-内射与 $C$-$fp_n$-平坦模类，研究其维数、Foxby 等价、预包络与覆盖的存在性，并在几乎优扩张下探讨其交换性质。

模的局部化的同调不变量的渐近行为

Asymptotic behavior of homological invariants of localizations of modules

本文研究交换诺特环上模的商模局部化的同调不变量，证明其内射维数、投射维数、Bass 数与 Betti 数在大整数 n 下稳定或呈多项式增长，且该性质不依赖于素理想的选取。

On $L_{n}$-Injective Modules and $L_{n}$-Injective Dimensions

Let $R$ be a ring, and $n$ a fixed nonnegative integer. An $R$-module $W$ is called $L_{n}$-injective if ${\rm Ext}_{R}^{1}(M,W)=0$ for any $R$-module $M$ with flat dimension at most $n$. In this paper, we prove first that ($\mathcal{F}_{n},\mathcal{L}_{n}$) is a complete hereditary cotorsion theory, where $\mathcal{F}_n$ (resp. $\mathcal{L}_n$) denotes the class of all $R$-modules with flat dimension at most $n$ (resp. $L_{n}$-injective $R$-modules). Then we introduce the $L_{n}$-injective dimension of a module and $L_n$-global dimension of a ring. Finally, over rings with weak global dimensi…

单侧Gorenstein环

One-sided Gorenstein rings

本文证明Iwanaga--Gorenstein环的若干特征可推广至单侧情形，无需诺特假设，并在Gorenstein整体维数、模维数上确界及有限维数间建立新关系。

凝聚环上Gorenstein平坦模的直积

Direct products of Gorenstein flat modules over coherent rings

本文构造了零Krull维数的交换凝聚局部代数及相关模，否定回答了右凝聚环上Gorenstein平坦模是否对直积封闭的问题，给出了相关等价条件并揭示了抽象初等类的性质。

群扩张的Gorenstein同调维数

Gorenstein homological dimension of group extensions

该研究针对交换环k上FP_∞型群，证明其三类自然维数一致，建立Fel'dman定理的Gorenstein同调类似结果等，拓展了群同调理论相关结论。

环的 n-平凡扩张的 Gorenstein 同调性质

Gorenstein homological properties of n-trivial extensions of rings

本文研究环的 n-平凡扩张上的 Gorenstein 投射、内射和平坦模，给出投射模的等价刻画，并应用于张量环。

张量环上的Gorenstein同调模

Gorenstein homological modules over tensor rings

本文研究了张量环上Gorenstein同调模的性质，提出了Gorenstein投射模的特征，并探讨了其在环扩展和Morita上下文环中的应用。

环的经典与Gorenstein同调不变量

On classical and Gorenstein homological invariants of rings

本文证明任意环R上平坦左R-模的投射维数上确界等于Gorenstein平坦左R-模的Gorenstein投射维数上确界，由此得到左n-完全环的新刻画，并引入经典同调不变量的Gorenstein类比，研究其关系，最后给出左弱n-Σ-余挠环的Gorenstein类刻画。

Homological dimension formulas for trivial extension algebras

Let $A= Λ\oplus C$ be a trivial extension algebra. The aim of this paper is to establish formulas for the projective dimension and the injective dimension for a certain class of $A$-modules which is expressed by using the derived functors $- \otimes^{\mathbb{L}}_ΛC$ and $\mathbb{R}\text{Hom}_Λ(C, -)$. Consequently, we obtain formulas for the global dimension of $A$, which gives a modern expression of the classical formula for the global dimension by Palmer-Roos and Löfwall that is written in complicated classical derived functors. The main application of the formulas is to give a necessary and…

Modules with cosupport and injective functors

Several authors have studied the filtered colimit closure lim(B) of a class B of finitely presented modules. Lenzing called lim(B) the category of modules with support in B, and proved that it is equivalent to the category of flat objects in the functor category (B^{op},Ab). In this paper, we study the category (Mod-R)^B of modules with cosupport in B. We show that (Mod-R)^B is equivalent to the category of injective objects in (B,Ab), and thus recover a classical result by Jensen-Lenzing on pure injective modules. Works of Angeleri-Hugel, Enochs, Krause, Rada, and Saorin make it easy to discu…

模的 Harmanci 内射性

Harmanci Injectivity of Modules

本文研究 Harmanci 内射模类，给出其刻画与性质，证明其构成包络并与特征模为 Matlis 内射的模类形成完美余挠理论，并探讨环的内射包络何时平坦。

大族无限群上的Gorenstein模与维数

Gorenstein modules and dimension over large families of infinite groups

该研究刻画了大族无限群的群代数上的三类Gorenstein模，证明弱Gorenstein模与对应Gorenstein模的等价性，得到有限Gorenstein弱整体维数交换环上的相关结论，并确定了特定群的Gorenstein同调维数。

FP-injective and weakly quasi-Frobenius rings

The classes of FP-injective and weakly quasi-Frobenius rings are investigated. The properties for both classes of rings are closely linked with embedding of finitely presented modules in fp-flat and free modules respectively. Using these properties, we describe the classes of coherent CF and FGF-rings. Moreover, it is proved that the group ring R(G) is FP-injective (resp. weakly quasi-Frobenius) if and only if the ring R is FP-injective (resp. weakly quasi-Frobenius) and the group G is locally finite.

Gorenstein injective envelopes and covers over two sided noetherian rings

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.